Incomplete Information Processing: A Solution to the Forward Discount Puzzle 1

Transcription

1 Incomplete Information Processing: A Solution to the Forward Discount Puzzle 1 Philippe Bacchetta Study Center Gerzensee University of Lausanne Swiss Finance Institute & CEPR Eric van Wincoop University of Virginia NBER September 1, This is a revised version of a paper previously circulated under the title Rational Inattention: A Solution to the Forward Discount Puzzle. We would like to thank Gianluca Benigno, Eric Fisher, Richard Lyons, Nelson Mark, Michael Melvin, Michael Moore, Jaume Ventura, and participants at an IFM NBER meeting, an RTN International Capital Markets meeting, the 2006 Annual Pacific Basin Conference at the Federal Reserve Bank of San Francisco and seminar participants at the Board of Governors, Cornell, Illinois, Ohio State, Pompeu Fabra, Zurich, St. Gallen, and the Hong Kong Monetary Authority. We also thank Elmar Mertens for outstanding research assistance and Michael Sager for many useful discussions about the operation of the foreign exchange market. van Wincoop acknowledges financial support from the Bankard Fund for Political Economy and the Hong Kong Institute for Monetary Research. Bacchetta acknowledges financial support by the National Centre of Competence in Research Financial Valuation and Risk Management (NCCR FINRISK).

2 Abstract The uncovered interest rate parity equation is the cornerstone of most models in international macro. However, this equation does not hold empirically since the forward discount, or interest rate differential, is negatively related to the subsequent change in the exchange rate. This forward discount puzzle implies that excess returns on foreign currency investments are predictable. Motivated by the fact that even today only a tiny fraction of foreign currency holdings are actively managed, we investigate to what extent incomplete information processing can explain this puzzle. Two types of incompleteness are considered: infrequent and partial information processing. We calibrate a two-country general equilibrium model to the data and show that incomplete information processing can fully match the empirical evidence. It can also account for several related empirical phenomena, including that of delayed overshooting. We also show that incomplete information processing is optimal. Predictability is largely overshadowed by uncertainty about future exchange rates, so that the welfare gain from actively managing foreign exchange positions is small and easily outweighed by a small cost of active portfolio management.

3 1 Introduction One of the best established and most resilient puzzles in international finance is the forward discount puzzle. 1 Fama (1984) illuminated the problem with a regression of the monthly change in the exchange rate on the preceding one-month forward premium. The uncovered interest rate parity equation, which is the cornerstone of many models in international macro, implies a coefficient of one. But surprisingly Fama found a negative coefficient for each of nine different currencies. A currency whose interest rate is high tends to appreciate. This implies that high interest rate currencies have predictably positive excess returns. The relationship between excess returns and interest rate differentials is illustrated in Table 1 for five currencies against the U.S. dollar. A regression of the quarterly excess return on a foreign currency on the difference between the U.S. and foreign interest rate yields coefficients ranging from -1.5 to Moreover, as documented below, interest rate differentials continue to negatively predict the excess returns five to ten quarters ahead. Most models assume that investors incorporate instantaneously all new information in their portfolio decisions. To explain the forward premium puzzle, we depart from this assumption. Portfolio decisions are usually not made on a continuous basis. While there now exists an industry that actively manages foreign exchange positions of investors, it only developed in the late 1980s and still manages only a tiny fraction of cross border financial holdings. 3 Outside this industry there is little active currency management over horizons relevant to medium-term 1 For surveys see Lewis (1995), Engel (1996), or Sarno (2005). Some of the more recent contributions include Backus, Foresi and Telmer (2001), Beakert, Hodrick and Marshall (1997), Chaboud and Wright (2005), Chinn and Meredith (2005), Chinn and Frankel (2002), Fisher (2006), Flood and Rose (2002), Gourinchas and Tornell (2004), Mark and Wu (1998), Sarno, Valente and Leon (2006) and Verdelhan (2005). 2 While there are potential statistical problems in these predictability regressions (mainly small sample bias and bias caused by the persistence of the forward discount), these problems usually can only explain a part of the total bias. See, for example, Stambaugh (1999), Campbell and Yogo (2006), or Liu and Maynard (2005). 3 It consists of hedge funds exploiting forward discount bias and financial institutions that provide such services to individual clients. The latter include currency overlay managers, commodity trading advisors and leveraged funds offered by established asset management firms. See Sager and Taylor (2006) for a recent description of the foreign exchange market. 1

4 excess return predictability. Banks conduct extensive intraday trade, but hold virtually no overnight positions. 4 Mutual funds do not actively exploit excess returns on foreign investment since they only trade within a certain asset class and cannot freely switch between domestic and foreign assets. Finally, Lyons (2001) points outthatmostlargefinancial institutions do not even devote their own proprietary capital to currency strategies based on the forward discount bias. Thus, any close examination of the functioning of the foreign exchange market leads one to conclude that information is incorporated incompletely into portfolio decisions. Incomplete information processing can take two different forms: (i) infrequent information processing, where investors make portfolio decisions infrequently, and (ii) partial information processing, where investors use only a subset of all available information. Anecdotal evidence suggests that even the most active traders use only very limited information to predict future exchange rates. Many simply assume that the exchange rate follows a random walk, given the well-known difficulty in doing much better than that. 5 We examine the impact of incomplete information processing in a simple twocountry general equilibrium model that is calibrated to data for the five currencies in Table 1. Agents are fully rational, but face a cost of making active portfolio decisions that we take as given. While fees charged for active management of foreign exchange positions tend to be substantial, we find that even for a quite small cost, most investors do not find it in their interest to actively exploit all available information. 6 Such a framework can account for both the sign and size 4 Two thirds of trade in the foreign exchange market is done among banks that are foreign exchange dealers (BIS, 2004). But since they hold little foreign exchange overnight, the huge intraday trading volume in the forex market is mostly irrelevant for medium-term excess return predictability. Chaboud and Wright (2005) show that there is actually little predictability with intraday data. 5 SeeMeeseandRogoff (1983) and more recently Cheung et al. (2005). 6 There is no established statistic on management fees. But everything indicates that active portfolio managers, such as hedge funds, charge fees that are often well above 2% of invested funds. An interesting question is why these fees are high. They are likely to reflect three elements: (i) the costs associated with collecting and processing information, computing the optimal portfolio, and attracting and distributing funds, (ii) profit margins due to their financial expertise and product differentiation and (iii) a profit sharing component intended to deflect agency and monitoring costs. There exists a substantial literature investigating the compensation of porftolio managers. See for example Berk and Green (2005) or Dybvig, Farnsworth and 2

5 of forward discount bias illustrated in Table 1. There are two distinct features that are surprising in the forward discount anomaly. The first aspect is the consistent sign of the bias. Why would the excess return be high for currencies whose interest rate is relatively high? This can be explained by infrequent information processing by investors. Froot and Thaler (1990) and Lyons (2001) have informally argued that models where some agents are slow in responding to new information may explain the forward discount puzzle. The argument is quite simple. An increase in the interest rate of a particular currency will lead to an increase in demand for that currency and therefore an appreciation of the currency. But when investors make infrequent portfolio decisions, they will continue to buy the currency as time goes on. 7 This can cause a continuing appreciation of the currency, consistent with the evidence documented by Fama (1984) that an increase in the interest rate leads to a subsequent appreciation. It also implies that a higher interest rate raises the expected excess return of the currency. Infrequent information processing can also explain the dynamic response of currency depreciation, or excess returns, to changes in interest rates. Interestingly, predictability is not restricted to horizons of a month or a quarter: the forward discount at time t can also predict excess returns at future dates. This feature is typically overlooked in the literature. Consider a regression of a future three-month excess return q t+k,fromt+k 1tot+k, on the current interest rate differential i t i t. Figure 1 shows the evidence for the five countries in Table 1, where k increases from 1 to 30. There is significant predictability with a negative sign for five to ten quarters. Over longer horizons, however, the slope coefficient becomes insignificant or even positive. This is consistent with findings that uncovered interest parity holdsbetteratlongerhorizons. 8 The persistence in the predictability of excess returns is related to the phenomenon of delayed overshooting. Eichenbaum and Evans (1995) first documented that after an interest rate increase, a currency continues to appreciate for another 8 to 12 quarters before it starts to depreciate. 9 Carpenter (2004) and references therein. 7 This is consistent with the evidence in Froot, O Connell, and Seasholes (2001), who show that cross-country equity flows react with lags to a change in returns, while the contemporaneous reaction is muted. 8 See for example Chinn and Meredith (2005), Boudoukh et al. (2005), or Chinn (2006). 9 Gourinchas and Tornell (2004) explain both predictability and delayed overshooting with distorted beliefs on the interest rate process. 3

6 As pointed out above, this is exactly what one expects to happen when investors make infrequent portfolio decisions. The second surprising aspect of the forward premium puzzle is that investors do not exploit the predictability of excess returns. The standard explanation is that an excess return reflects a risk premium. But many surveys written on the forward discount puzzle have concluded that explanations for the forward discount puzzle related to time-varying risk premia have all fallen short. 10 Our analysis shows that, given the high risk involved, a small asset management cost discourages investors from exploiting the predictability. This risk is illustrated in Figure 2, which shows for one currency, the DM/$, a scatter plot of the excess return on DM against the U.S. minus German interest rate differential. The negative slope of the regression line represents predictability. It is clear though that predictability is largely overshadowed by risk. 11 Thismeansthatformanyinvestorsitissimply not worthwhile to actively trade on excess return predictability. Even for those who do actively trade on the excess return predictability, the high risk limits the positions they will take. We will show in the context of the model that a small fraction of financial wealth actively devoted to forward bias trade will not unravel theimpactofinfrequentdecisionmaking. It is the combination of infrequent and partial information processing that is key to our results. Infrequent information processing by itself leads to predictability of the right sign, but does not fully match the data quantitatively. On the other hand, partial information processing by itself leads to virtually no predictability or predictability of the wrong sign. It is the combination of the two perspectives that closely matches the data. The distinction between partial use of information and infrequent information processing is also found in the recent literature on rational inattention (or inattentiveness) in macro models. One strand of the literature, based on Sims (1998, 2003), considers continuous but partial information processing due to (Shannon) capacity constraints. In another strand of the literature, e.g., Mankiw and Reis (2002), there are time-dependent decision rules, where in- 10 See Lewis (1995) or Engel (1996). Recently Verdelhan (2005) has more success based on a model with time-varying risk aversion due to habit formation. On the other hand, Burnside et al. (2006) find that excess returns are uncorrelated with risk factors. 11 More formally, this is reflected in the low R 2 for excess return regressions in Table 1, which is on average

7 formation is processed infrequently. 12 Although the two types of approaches are related, they have a different impact in an asset pricing context. Our theoretical analysis is also related to recent developments in the stock market literature. 13 On the one hand, several studies show how asset allocation is affected by predictability. 14 On the other hand, some recent papers examine the impact of infrequent portfolio decisions due to limited attention in asset markets. 15 However, the literature has not linked predictability with infrequent trading: those papers that examine the impact of predictability assume it exogenous, while papers that examine infrequent portfolio decisions do not examine its impact on asset prices. Our paper departs from the existing literature by incorporating both predictability and infrequent portfolio decisions and by showing that the latter can cause the former. Our methodological contribution to the literature is to solve endogenously for an asset price in a model with time-varying expected returns. The remainder of the paper is organized as follows. Section 2 describes a twocountry general equilibrium model where all investors make infrequent portfolio decisions. The model is calibrated to data for the five currencies in Table 1. Section 3 discusses the implications of the model for the forward discount and delayed overshooting puzzles. It also considers an extension of the model to partial information processing and to investors that actively manage their portfolio each period. Section 4 relates our analysis to the existing literature on the forward 12 There is a growing literature in macroeconomics based on rational inattention, in particular in the context of price setting by firms and consumption decisions by households. Examples of papers where agents process partial information due to information capacity constraints are Sims (1998, 2003) and Mackowiak and Wiederholt (2005). Examples of papers where agents process information infrequently due to explicit information processing costs are Begg and Imperato (2001), Bonomo and de Carvalho (2004), Moscarini (2004), and Reis (2006a,b). Carroll (2003), Dupor and Tsuruga (2005) and Mankiw and Reis (2002) assume exogenously that new information arrives, and is processed, at a certain rate (either with a fixed probability or at fixed intervals). 13 Evidence of excess return predictability has been extensively documented for stock and bond markets (e.g. see Cochrane, 1999). 14 See for example Kandel and Stambaugh (1996), Campbell and Viceira (1999), or Barberis (2000). 15 Duffie and Sun (1990), Lynch (1996), and Gabaix and Laibson (2002) have all developed models where investors make infrequent portfolio decisions because of a fixed cost of information collection and decision making. 5

8 discount puzzle. Section 5 concludes. 2 A Model of Infrequent Decision Making This section presents a model of the foreign exchange market where investors make infrequent portfolio decisions. First the basic structure of the model and the solution method are described. We then discuss under what cost of active portfolio management it is optimal for all investors to make infrequent portfolio decisions. Some technical details are covered in the Appendix, with a Technical Appendix available on request providing full technical detail. 2.1 Model s Description Basic Setup We develop a one good, two-country, dynamic general equilibrium model. The overall approach is to keep the model as simple as possible while retaining the key ingredients needed to highlight the role of infrequent decision making. There are overlapping generations (OLG) of investors who each live T + 1 periods and derive utility from end-of-life wealth. Each period a total of n new investors are born, endowedwithoneunitofthegoodthatcanbeinvestedinassetsdescribedbelow. The infrequent decision making is modeled by assuming that investors make only one portfolio decision when born for the next T periods. The threshold portfolio management cost under which it is indeed optimal to make infrequent portfolio decisions is derived below. This OLG setup is easier to work with than the alternative where agents have infinite horizons and make portfolio decisions every T periods. In that case optimal saving-consumption decisions have to be solved for as well and depend on assumptions made about the frequency of those decisions. We have abstracted from saving decisions by assuming that agents derive utility from end-of-life wealth. This allows us to focus squarely on portfolio decisions. 16 We want to emphasize though 16 An infinite horizon setup would complicate matters in other ways as well. The optimal portfolio would be hard to compute since it depends on a hedge against changes in expected returns T periods from now. One would also need to introduce additional features to induce stationarity of the wealth distribution. 6

9 that while an infinite horizon setup is more complicated, the mechanisms at work are similar to those in our simpler OLG framework. The crucial element is that information is incorporated gradually into portfolio decisions because only a limited fraction of agents make new portfolio decisions each period. It is of little relevance for what follows whether this new information is incorporated by a new generation, as in the OLG model, or by a subset of infinitely-lived investors. The model contains one good and three assets. In the goods market purchasing power parity holds: p t = s t + p t,wherep t is the log-price level of the good in the Home country and s t the log of the nominal exchange rate. Foreign country variables are indicated with a star. The three assets are one-period nominal bonds in both currencies issued by the respective governments and a risk-free technology with real return r. 17 Bonds are in fixed supply in the respective currencies. 18 We first describe the monetary policy rules adopted by central banks, then optimal portfolio choice, and finally asset market clearing Monetary Policy The Home country central bank commits to a constant price level. This implies zero Home inflation, so that the Home nominal interest rate is i t = r. Theforeign interest rate is random, i t = u t where u t = ρu t 1 + ε u t ε u t N(0, σ 2 u) (1) The error term captures foreign monetary policy innovations. The forward discount is: fd t i t i t = u t + r (2) These assumptions imply that there are in essence only two assets, one with a risk-free real return r and one with a stochastic real return. The latter is Foreign bonds, which has a real return of s t+1 s t + i t. This setup leads to much simpler 17 This is necessary to tie down the real interest rate since the model does not contain saving and investment decisions. 18 One can think of the governments that issue the bonds as owning claims on the riskfree technology whose proceeds are sufficient to pay the interest on the debt. The remainder is thrown in the water or spent on public goods that have no effect on the marginal utility from private consumption. 7

10 portfolios than one would get under symmetric monetary policy rules, in which casetherealreturnonhomeandforeignbondswouldbothbestochastic Portfolio Choice Since PPP holds, Foreign and Home investors face the same real returns and therefore choose the same portfolio. They have constant relative risk-aversion preferences over end-of-life consumption, with a rate of relative risk-aversion of γ. Investors born at time t maximize E t W 1 γ t+t /(1 γ), where W t+t is end-of-life financial wealth that will be consumed. Investors make only one portfolio decision when born, investing a fraction b I t in Foreign bonds. 20 End of life wealth is then W t+t = TY k=1 R p t+k (3) where R p t+k is the gross investment return from t + k 1tot + k, R p t+k =(1 b I t )e i t+k 1 + b I t e s t+k s t+k 1 +i t+k 1 (4) In order to solve for optimal portfolios, a second order approximation of log portfolio returns is adopted. 21 Define q t+k = s t+k s t+k 1 + i t+k 1 i t+k 1 as the excess return on Foreign bonds from t + k 1tot + k and q t,t+t = q t q t+t as the cumulative excess return from t to t + T. Appendix A.1 shows that the optimal portfolio rule is b I t = b I + E tq t,t+t (5) γσi 2 where b I is a constant and σi 2 is defined as σ 2 I = Ã 1 1 γ! var t (q t,t+t )+ 1 γ TX k=1 var t (q t+k ) (6) 19 Without having to introduce nominal rigidities, from the point of view of the Home country it also captures the fact that exchange rate risk is far more substantial than inflation risk. 20 The portfolio share is held constant for T periods, which fits reality better than investors deciding on an entire path of portfolio shares for the next T periods. 21 The objective function is maximized after replacing the log portfolio returns by their second order approximation. An alternative solution method is to start from the first order condition for portfolio choice and then substitute a first order approximation of the log portfolio return. This gives exactly the same solution. The latter is the approach adopted by Engel and Matsumoto (2005) to solve for optimal portfolios in a general equilibrium model with home bias. 8

11 The optimal portfolio therefore depends on the expected excess return over the next T periods, with less aggressive portfolio choices made when either agents are more risk averse or there is more uncertainty about future returns Liquidity Traders There is another group of investors referred to as liquidity traders. In the noisy rational expectations literature in finance it is common to introduce exogenous noise or liquidity traders since this noise prevents the asset price from revealing the aggregate of private information. Here there is no private information, but exogenous liquidity traders are introduced in order to match two key features of exchange rate data. 22 First, it is important to match the observed exchange rate volatility in the data since it affects optimal portfolios through uncertainty about future excess returns. Interest rate shocks alone are not nearly sufficient in this regard and it would also violate extensive evidence that observed exchange rate volatility is largely disconnected from observed macro fundamentals. 23 Second, it is important to match the well-known stylized fact that exchange rates behave close to a random walk. This is of clear relevance in the decision about whether to actively manage the portfolio or not. If there were large predictable components to exchange rate changes, the gain from active portfolio management would obviously be larger. Interest rate shocks alone do not generate an exchange rate that is close to a random walk. The real value of Foreign bond investments by liquidity traders at time t is ( x + x t ) W,where W is aggregate steady state financial wealth and x t follows the process: x t = C(L)ε x t =(c 1 + c 2 L + c 3 L )ε x t ε x t N(0, σ 2 x) (7) The magnitude of the shocks is chosen to match observed exchange rate volatility 22 The exogenous noise that is generated by liquidity traders can also be modeled endogenously, without any implications for the results. See Bacchetta and van Wincoop (2006). 23 A substantial literature has confirmed the initial findings by Meese and Rogoff (1983) that observed macro fundamentals explain very little of exchange rate volatility for horizons up to 1 or 2 years. Lyons (2001) has called this the exchange rate determination puzzle. Bacchetta and van Wincoop (2004, 2006) show that in the presence of heterogenous information even small liquidity shocks can have a large effect on exchange rates movements, so that exchange rates are disconnected from macroeconomic fundamentals. 9

12 and the polynomial C(L) such that in equilibrium the exchange rate is close to a random walk. We will return to this below when discussing the solution method. It is important to note that liquidity trade shocks do not directly contribute to excess return predictability associated with the forward discount. The reason is that we do not allow these shocks to affect interest rates, either directly or indirectly Market Clearing The last model equation is the Foreign bond market clearing condition. There is a fixed supply B of Foreign bonds in the Foreign currency. The real supply of Foreign bonds is Be p t = Be s t, where the Home price level is normalized at 1 (so that p t = 0). Investors are born with an endowment of one, but their wealth accumulates over time. Let Wt k,t I be the wealth at time t foraninvestorborn at t k. This is equal to the product of total returns over the past k periods, Wt k,t I = Q k j=1 R p t k+j. The market clearing condition for Foreign bonds is then n TX k=1 b I t k+1w I t k+1,t +(x + x t ) W = Be s t (8) The constant x is set such that the steady state supply of Foreign bonds relative to total financial wealth, Be s / W,isequaltob, which is set exogenously. Without loss of generality, the nominal supply B is such that this holds for a zero steady state log exchange rate: s =0. A couple of points are worth making about the market clearing condition. In order for the frequency of portfolio decisions to matter, portfolios should adjust in equilibrium after an interest rate shock. If supply is entirely fixed in domestic currency and no other agents are willing to take the other side of the transaction, portfolios will not change in equilibrium. In our model supply adjusts because it depends on the exchange rate. An increased demand for Foreign bonds raises the supply of Foreign bonds through a depreciation of the Home currency (s t rises). This effect is partially offset by a wealth effect for agents who are not making 24 In a previous version of the paper, we assumed an interest rate rule reacting to the exchange rate. In that context, liquidity trade contributes to the forward bias puzzle since liquidity shocks are correlated with the interest rate. For this impact to be large, however, the interest rate must be very sensitive to the exchange rate. This is the mechanism emphasized by McCallum (1994). 10

13 new portfolio decisions. An increase in Foreign bond returns due to a rise in s t raises their wealth and therefore increases their demand for Foreign bonds at the constant portfolio shares chosen when born. So even when traders do not make new portfolio decisions, they still conduct some trade associated with portfolio rebalancing Solving the Model We now briefly outline the solution method, leaving details to Appendix A.2 and the Technical Appendix. The first step is to linearize the market clearing condition for Foreign bonds around the point where the log exchange rate and asset returns are zero and portfolio shares are equal to their steady-state values. After substituting the optimal portfolios (5) into the market equilibrium condition, the equilibrium exchange rate can be derived. Start with the following conjecture for the equilibrium exchange rate: s t = A(L)ε u t + B(L)ε x t (9) where A(L) =a 1 + a 2 L +... and B(L) =b 1 + b 2 L +... are infinite lag polynomials. Conditional on this conjectured exchange rate equation, compute excess returns as well as their first and second moments that enter into the optimal portfolios. One can then solve for the parameters of the polynomials by imposing the linearized bond market equilibrium condition. ButratherthansolvingforA(L) andb(l) given the model and the process for interest rate and liquidity demand shocks, we solve instead for A(L), b 1 and C(L) such the that (i) the Foreign bond market equilibrium condition is satisfied and (ii) ˆx t = B(L)² x t follows an AR process: ˆx t = ρ xˆx t 1 + b 1 ² x t (10) The latter implies b k = ρ k 1 x b 1 for k > 1. Rather than taking the process of liquidity demand shocks as given, it is chosen such that the impact of these shocks on the exchange rate follows an AR process. By setting the AR coefficient ρ x close to 1, the exchange rate then becomes close to a random walk. As discussed in the Appendix, b 1 and A(L) can be solved jointly. After that, the parameters of the polynomial C(L) follow immediately from the market clearing condition. But C(L) is not consequential for the rest of the analysis. Since the 11

14 polynomial A(L) has an infinite number of parameters, and solving it jointly with b 1 therefore requires solving an infinite number of non-linear equations, the polynomial A(L) is truncated after T lags. We set a k =0fork> T and solve b 1,a 1,..,a T from T + 1 non-linear equations. Since interest rate shocks are temporary, their impact on the exchange rate dies out anyway, making this approximation very precise for large T. Inpracticeweset T so large that increasing it any further has no effect on the results. 2.2 On the Optimality of Infrequent Decision Making Under what circumstances is the passive portfolio management strategy followed by all traders in the model optimal? There is a trade-off between the higher expected returns under active portfolio management and the cost involved. Assume that the cost of active portfolio management is a fraction τ of wealth per period. 25 The question then is how large τ needs to be for it to be optimal for all traders to make decisions infrequently. We will refer to the level of τ where expected utility is the same under active and passive portfolio management strategies as the threshold cost. Aslongasτ is above this threshold, it is optimal for traders to make infrequent portfolio decisions. For now, all traders face the same cost τ. In the next section we will also consider a case where the cost τ differs across agents, so that it is possible that some choose to actively manage their portfolio while others make infrequent portfolio decisions. In order to determine the threshold cost, we must consider the alternative where traders make portfolio decisions each period. 26 An investor with an actively managed portfolio must solve a more complicated multi-period portfolio decision problem. Since equilibrium expected returns are time varying, the optimal dynamic portfolio contains a hedge against changes in future expected returns. A technical contribution of the paper is to derive an explicit analytical solution to the multi-period portfolio decision problem with time-varying expected returns. Here we briefly describe the method, leaving the details to Appendix A.1 and the Technical Appendix. 25 It actually makes little difference whether this cost is a constant or proportional to wealth since initial wealth is 1 and the product of τ and the subsequent change in wealth is second order. 26 We will abstract from scenarios where agents make portfolio decisions at intervals between one and T. 12

15 First, conjecture that the value function at time t + k (k =0,..,T)ofanagent born at time t is V t+k = e Y 0 t+k H ky t+k (1 τ) (1 γ)(t k) W 1 γ t+k /(1 γ) (11) Here W t+k is wealth at t +k, H k is a matrix and Y t+k is the state space. The latter consists of Y t+k =(ε u t+k,.., ε u t+k+1 T, ˆx t, 1) 0. Since in principle the state space is infinitely long, for tractability reasons it is truncated after T periods (with T very large), similar to the exchange rate solution. The key conjecture is that the term in the exponential of the value function is quadratic in the state space. At time t + k the optimal portfolio is chosen by maximizing E t+k V t+k+1.first substitute W t+k+1 =(1 τ)w t+k e rp t+k+1 into the expression for Vt+k+1,wherer p t+k+1 is a second order approximation of the log portfolio return from t + k to t + k +1. Then maximize with respect to the portfolio at t + k. It is shown that V t+k = E t V t+k+1 indeed takes the conjectured form in (11). Starting with the known value function at t + T, V t+t = W 1 γ t+t /(1 γ), which corresponds to H T =0,the value function for earlier periods is solved with backward induction, until the value function at time t is computed. The solution to this portfolio problem yields the following optimal portfolio share invested in Foreign bonds at time t + k foraninvestorbornattimet: b F t,t+k = b F E t+k (q t+k+1 ) (k)+ (γ 1)ˆσ F 2 (k)+σf 2 + D k Y t+k (12) The first term, b F (k), is a constant. The second term depends on the expected excess return over the next period. In the denominator σf 2 = var t (q t+1 ). The term ˆσ F 2 (k) isdefined in the Appendix but in practice is very close to var t (q t+1 ), so that the denominator is close to γvar t (q t+1 ). The third term captures a hedge against changes in future expected returns. D k is a vector of constant terms, so this term is linear in the state space. Assume that each new generation consists of n F agents who make frequent portfolio decisions, actively managing their portfolio each period, and n I agents who make infrequent portfolio decisions, with n = n I +n F. The market equilibrium condition then becomes X T n F k=1 b F t k+1,tw F t k+1,t + n I T X k=1 b I t k+1w I t k+1,t +( x + x t ) W = Be s t (13) 13

16 where Wt k+1,t F is the wealth at time t of agents born at time t k +1 who actively manage their portfolio. In section 3.3 we will consider the case where the fraction of agents that actively manages their portfolio is positive. For now we focus on the case where it is optimal for all agents to make infrequent portfolio decisions. In that case n F =0 in equilibrium and n I = n. This is the case as long as the cost of active portfolio management is higher than the threshold cost. The threshold cost τ is determined such that the expected utility of an investor making frequent portfolio decisions is the same as that of an investor making infrequent portfolio decisions. Since each investor starts with wealth equal to 1, the value function at birth for an investor making frequent portfolio decisions is e Y t 0H 0Y t (1 τ) (1 γ)t /(1 γ). For an investor making only one portfolio decision for T periods, the time t value function is V t = E t W 1 γ t+t /(1 γ). After substituting W t+t = e rp t rp t+t, maximization with respect to b I t yields the optimal portfolio (12) and a time t value function that takes the form e Y t 0HY t /(1 γ). When born, investors need to decide whether to actively manage their portfolio before observing the state Y t. 27 We therefore compare the unconditional expectation of the time t value functions for the two strategies, where the expectation is with respect to the unconditional distribution of Y t. The threshold cost τ is such that expected utility is the same under both strategies. 2.3 Parameterization The model is calibrated to data for the five currencies on which Table 1 and Figure 1 are based. Consistent with the quarterly excess returns in Table 1 and Figure 1, a period is set equal to one quarter. The AR process for the forward discount, and therefore u t, is estimated for the countries and sample period corresponding to the excess return regression reported in Table The parameters ρ u and σ u are set equal to the average across the countries of the estimated processes. This 27 In a more realistic framework where agents have infinite lives and make portfolio decisions every T periods, this corresponds to agents deciding on the frequency of portfolio decisions before observing future states when portfolio decisions are actually made. In other words, it corresponds to a time-dependent decision rule. 28 We use three-month Euro-market interest rates from Datastream between December 1978 and December

17 yields ρ u =0.8 andσ u = The process for the supply x t = C(L)² x t cannot be observed directly. As discussed above, this process is chosen to match observed exchange rate volatility and the near-random walk behavior of exchange rates. To be precise, σ x is set such that the standard deviation of s t+1 s t in the model is equal to the average standard deviation of the one quarter change in the log exchange rate for the five currencies and time period of the excess return regression reported in Table 1. The average standard deviation is The polynomial C(L) is chosen such that ˆx t follows anarprocessasin(10)witharcoefficient ρ x =0.99. This means that the exchange rate is close to a random walk since liquidity demand shocks dominate exchange rate volatility. In the benchmark parameterization we set T = 8. Thisimpliesthatagents make one portfolio decision in two years, so that half of the agents change their portfolio during a particular year. While it is hard to calibrate this precisely to the data for the foreign exchange market, it corresponds well to evidence for the stock market. The Investment Company Institute (2002) reports that only 40% of U.S. investors change their stock or mutual fund portfolios during any particular year. 29 Trade in the foreign exchange market is closely tied to international trade in stocks, bonds and other assets. Setting T = 8 also corresponds well to evidence reported by Parker and Julliard (2005) and Jagannathan and Wang (2005) that Euler equations for asset pricing better fit the data when returns are measured over longer horizons of one to three years. In section 4 we will further discuss that evidence and its connection to our model. The final two parameters are b and γ. 30 We set b =0.5, corresponding to a two-country setup with half of the assets supplied by the US and the other half bytherestoftheworld. Therateofrelativeriskaversionissetat10. Thisisin the upper range of what Mehra and Prescott (1985) found to be consistent with estimates from micro studies, but consistent with more recent estimates by Bansal and Yaron (2004) and Vissing-Jorgenson and Attanasio (2003). 31 Arisk-aversion 29 For a discussion of evidence on infrequent trading see Bilias et al. (2005) and Vissing- Jorgenson (2004). 30 There is also the truncation parameter T used in the solution method, which is set at 60 quarters. Increasing it further does not affect the results. 31 The estimates in Bansal and Yaron (2004) are based on a general equilibrium model that can explain several well known asset pricing puzzles. The estimates in Vissing-Jorgenson and 15

18 of 10 also reduces the well known extreme sensitivity of portfolios to expected excess returns in this type of model Explaining the Forward Premium Puzzle We now examine the model s quantitative implications for excess return predictability. We will show that the model indeed generates such predictability. We first present the results in our benchmark case and provide the intuition on the mechanism leading to predictability. This is closely related to the phenomenon of delayed overshooting. We then report the threshold cost of active portfolio management such that investors are equally well off adopting a passive or active portfolio management strategy. The threshold cost is very small and certainly below any reasonable value of the true cost of active portfolio management. This justifies the infrequent decision making by all investors. While the model is able to explain excess return predictability, the regression coefficient in the excess return equation is smaller than in the data. This moment cannot be matched even for drastic changes in the values of γ and T. Drawing on a large number of small sample simulations, we also show that the difference with the data cannot be explained by small sample bias. However, we suggest two potential explanations that quantitatively line up the model to the data. First, when the model is simulated over 25-year samples, the range of regression estimates is wide. While the mean of the estimated predictability coefficients is less than in the data, a relatively large proportion of the regression coefficients are at least as large as in the data. Second, the estimated coefficient can be matched when we additionally assume partial information processing. Under partial information processing, investors either assume that the exchange rate is a random walk or only use the current interest rate differential to optimally predict future exchange rates. Attanasio (2003) are based on estimating Euler equations using consumption data for stock market participants. 32 Other ways to improve this feature include loss aversion preferences, habit formation preferences, parameter uncertainty, transaction costs, and portfolio benchmarking. 16

19 3.1 Benchmark Results Panel A of Figure 3 reports results when regressing excess returns q t+k on the forward discount fd t, similar to Figure 1. While standard models predict coefficients around the zero line, the model is able to generate negative coefficients for small values of k, followed by positive coefficients for larger k. The usual one-period ahead coefficient is equal to Panel B shows a scatter plot of interest rate differentials against subsequent one-period excess returns for one simulation of the model over 100 periods, which corresponds to 25 years. The scatter plot is similar to what is found in the data as shown in Figure 2. The interest differential predicts excess returns, but both in the model and the data the predictability is largely out-shadowed by risk. To summarize, the benchmark parameterization delivers significant excess return predictability in the right direction, but the extent of the predictability is less than in the data. In the data the regression coefficient is close to We will now give some intuition both for why this predictability occurs and what limits the extent of the predictability. Delayed Overshooting Figure 4 provides the key intuition behind our findings. Panel A shows the impulse response of the exchange rate to a one standard deviation decrease in the Foreign interest rate. It compares the benchmark case with the case where all investors make portfolio decisions each period. In the latter case there is standard overshooting, i.e., the lower Foreign interest rate causes an immediate appreciation of the Home currency, followed by a gradual depreciation. In that case the excess return predictability coefficient is close to zero (-0.014). 33 With infrequent portfolio decisions, however, there is delayed overshooting, consistent with the empirical findings of Eichenbaum and Evans (1995). The initial appreciation is now smaller, but the Home currency continues to appreciate in the following several quarters, after which it starts to gradually depreciate. The continued appreciation is a result of the delayed portfolio response of investors. Investors making portfolio decisions at the time the shock occurs sell Foreign bonds in response to the news of a lower Foreign interest rate. The next 33 The fact that it is not exactly zero is because the change in the exchange rate changes the real supply of the foreign asset, Be st, which has a small risk-premium effect. 17

20 period a different set of investors adjust their portfolio. They too will sell Foreign bonds in response to the lower interest rate, leading to a continued appreciation of the Home currency. The currency continues to appreciate for three quarters. Panel B shows the evolution of the forward discount and the excess return (computed using the path of the exchange rate in Panel A). The Figure shows that initially the drop in the excess return is larger than the rise in the forward discount. The reason is that the excess return s t+1 s t fd t decreases both because of the rise in the forward discount (lower Foreign interest rate) and the subsequent appreciation of the Home currency (negative change in the exchange rate). However, the Figure also shows that this is not long-lasting. Within three quarters the absolute decline in the excess return is less than the rise in the forward discount and at T = 8 quarters they both go in the same direction. This limits the magnitude of the negative excess return predictability coefficient. Related to that, the delayed overshooting in panel A only lasts 3 quarters, while Eichenbaum and Evans (1995) report empirical evidence indicating delayed overshooting lasting for two to three years. The reason why the delayed overshooting does not last longer than 3 quarters is that at that point investors start buying Foreign bonds again. Investors know that the Foreign interest rate will continue to be lower than the Home interest rate, but they also realize that eventually the Home currency will depreciate. The reason is that the investors who sold Foreign bonds at the time the shock happened will increase their holdings of Foreign bonds 8 quarters later when they adjust their portfolio again. 34 After all, the interest rate differential in favor of Home bonds is expected to be much smaller 8 quarters later. Three periods after the shock the expected depreciation of the Home currency over the next 8 quarters is sufficient to more than offset the expected interest differentials in favor of the Home bonds. Investors will then start buying Foreign bonds again, causing the Home currency to gradually depreciate. This of course assumes very careful forward looking behavior on the part of investors, processing all available information to predict the exchange rate two years into the future. This information processing capacity may be unrealistic, an issue to which we will turn below. 34 More precisely, and leading to the same outcome, they are replaced by a new generation that chooses a new portfolio. 18

21 Threshold Cost Following the method described in section 2.2, we find an annualized threshold cost of 0.27% of wealth. This means that it is indeed optimal for all investors to make infrequent portfolio decisions when the cost of active portfolio management is at least 0.27% of wealth. This number is far below fees charged by active portfolio managers, which do not even include additional agency and monitoring costs when delegating these decisions to fund managers and the transaction costs associated with frequent portfolio adjustments. 35 The reason that the threshold cost is so small is that there is so much uncertainty about future returns. Since the component of the exchange rate that depends on liquidity demand shocks is close to a random walk, virtually the entire predictability comes from interest rates. Panel B of Figure 3 illustrates that the predictability of excess returns by interest differentials is simply overwhelmed by uncertainty, as is the case in the data. Uncertainty impacts the threshold costs in two ways. First, the more uncertainty there is about the excess return the lower the welfare gain from a given portfolio response to expected excess returns under active portfolio management. Second, as shown in (12), the optimal portfolio response under active portfolio management is itself dampened significantly by uncertainty about the excess return. Therefore only a small cost of active portfolio management is sufficient for investors not to actively exploit predictability. Small Sample Results In order to allow for better comparison to results based on the data reported in Table 1 and Figure 1, we have also simulated a 25-year period for the model. Based on 1000 simulations of a 25-year period, the average excess return predictability is very close to the population moment of This means that there cannot be a systematic small sample bias. However, the excess return predictability varies quite considerably across simulations. This is consistent with empirical evidence that shows that the excess return coefficient tends to be unstable over time. Panel A of Figure 5 reports the frequency distribution. In 12% of cases the excess return predictability coefficient is less than -2. This means that the findings in the data 35 One dimension of transaction costs is price pressure. Burnside et al. (2006) argue that price pressure alone can explain why investors do not exploit excess return predictability. 19

22 are well within reach of the model. 36 Panel B reports the average of the regression coefficients of q t+k on fd t (k = 1,.., 30) for the 10% of simulations (100 simulations) generating the lowest coefficient for k = 1. The picture is very similar to Figure 1 based on the data. The average predictability coefficient is -2.6 for k = 1. It continues to be negative for about six quarters, dropping in absolute size as k increases. Alternative Parameterizations Table 2 presents results on the one-period ahead predictability coefficient and thethresholdcostforsomealternativevaluesoftherateofriskaversionγ and the frequency T of decision making. The excess return predictability coefficient is larger for higher values of γ and T, but not enough to match the data. Based on population moments generated by the model, it is not possible to match the empirical estimate of about 2.5 even when we substantially increase γ and T.It remains the case though that for a large range of parameters there is a substantial probability that the excess return predictability coefficient is less than -2 in simulations of a 25-year period. We also see that the threshold cost remains quite low for a wide range of parameters. It is highest for a low rate of risk-aversion of γ =1sinceagentsare then less averse to the risk associated with exploiting excess return predictability. 3.2 Partial Information Processing Although investors in the model make infrequent portfolio decisions, we have assumed that they use all available information when they make those decisions. In other words, investors have rational expectations and are able to determine the future behavior of other investors and the full path of future returns based on all information available today. As explained above, it is this forward looking behavior that leads investors to start buying Foreign bonds after three periods, which limits the extent of delayed overshooting. However, as shown in the rational inattention literature, in the presence of costly information processing it may be optimal for investors to only process partial 36 In contrast, the probability of this being the case is only 1.1% when all investors make portfolio decisions each period. 20